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Vol. 16 (2011), Paper no. 26, pages 792–829.
Journal URL
http://www.math.washington.edu/~ejpecp/
Quenched limits and fluctuations of the empirical measure
for plane rotators in random media
Eric Luçon∗eric.lucon@etu.upmc.fr
Abstract
The Kuramoto model has been introduced to describe synchronization phenomena observed in
groups of cells, individuals, circuits, etc. The model consists of N interacting oscillators on the
one dimensional sphere S1 , driven by independent Brownian Motions with constant drift chosen
at random. This quenched disorder is chosen independently for each oscillator according to
the same law µ. The behaviour of the system for large N can be understood via its empirical
measure: we prove here the convergence as N → ∞ of the quenched empirical measure to the
unique solution of coupled McKean-Vlasov equations, under weak assumptions on the disorder
µ and general hypotheses on the interaction. The main purpose of this work is to address the
issue of quenched fluctuations around this limit, motivated by the dynamical properties of the
disordered system for large but fixed N : hence, the main result of this paper is a quenched
Central Limit Theorem for the empirical measure. Whereas we observe a self-averaging for the
law of large numbers, this no longer holds for the corresponding central limit theorem: the
trajectories of the fluctuations process are sample-dependent.
Key words: Synchronization - quenched fluctuations - central limit theorem - disordered systems
- Kuramoto model.
AMS 2000 Subject Classification: Primary 60F05, 60K37, 82C44, 92D25.
Submitted to EJP on July 23, 2010, final version accepted March 17, 2011.
∗
Laboratoire de Probabilités et Modèles Aléatoires (CNRS U.M.R. 7599) and Université Paris 6 – Pierre et Marie Curie,
U.F.R. Mathematiques, Case 188, 4 place Jussieu, 75252 Paris cedex 05, France
792
1 Introduction
In this work, we study the fluctuations in the Kuramoto model, which is a particular case of interacting diffusions with a mean field Hamiltonian that depends on a random disorder. The Kuramoto
model was first introduced in the 70’s by Yoshiki Kuramoto ([17], see also [1] and references
therein) to describe the phenomenon of synchronization in biological or physical systems. More
precisely, the Kuramoto model is a particular case of a system of N oscillators (considered as elements of the one-dimensional sphere S1 := R/2πZ) solutions to the following SDE:
i,N
dx t
=
N
1X
N
b(x i,N , x j,N , ω j ) dt + c(x i,N , ωi ) dt + dB ti ,
t ∈ [0, T ], i = 1 . . . N ,
j=1
(1)
where T > 0 is a fixed (but arbitrary) time, b and c are smooth periodic functions. The Kuramoto
case corresponds to a sine interaction (b(x, y, ω) = K sin( y − x) and c(x, ω) = ω). This case which
has the particularity of being rotationally invariant (namely, if (x i,N )i is a solution of the Kuramoto
model, (x i,N + c)i , c a constant, is also a solution), will be referred to in this work as the sine-model.
The parameter K > 0 is the coupling strength and (ω j ) is a sequence of randomly chosen reals
(being i.i.d. realizations of a law µ). The sequence (ω j ) j is called disorder and represents the fact
that the behaviour of each rotator x j,N depends on its own local frequency ω j .
Due to the mean field character of (1), the behaviour of the system can be understood via its
empirical measure ν N , process with values in M1 (S1 × R), that is the set of probability measures on
oscillators and disorder:
∀(ω) ∈ RN , ∀t ∈ [0, T ],
N ,(ω)
νt
:=
N
1X
N
δ(x j,N ,ω ) ,
j=1
t
j
where (ω) = (ω j ) j≥1 is a fixed sequence of disorder in RN .
The purpose of this paper is to address the issue of both convergence and fluctuations of the empirical measure, as N → ∞ ; thus the main theorem of this paper (Theorem 2.10) concerns a Central
Limit Theorem in a quenched set-up (namely the quenched fluctuations of ν N around its limit).
Some heuristic results have been obtained in the physical literature ([1] and references therein)
concerning the convergence of the empirical measure, as N → ∞, to a time-dependent measure
(Pt ( dx, dω)) t∈[0,T ] , whose density w.r.t. Lebesgue measure at time t, q t (x, ω) is the solution of
a deterministic non-linear McKean-Vlasov equation (see Eq.(5)). It is well understood ([1], [11])
that crucial features of this equation are captured in the sine-model by order parameters r t and ψ t
defined by:
Z
r t e iψ t =
e i x q t (x, ω) dxµ( dω).
S1 ×R
1
The quantity r t captures in fact the degree of synchronization of a solution (the profile q t ≡ 2π
for example corresponds to r = 0 and represents a total lack of synchronization) and ψ t identifies
the center of synchronization: this is true and rather intuitive for unimodal profiles. Moreover
([22], see also [11], p.118) it turns out that if µ is symmetric, all the stationary solutions can be
parameterized (up to rotation) by the stationary version r of r t which must satisfy a fixed point
793
relation r = ΨK,µ (r), with ΨK,µ (·) an explicit function such that ΨK,µ (0) = 0. For K small, r = 0 is
the only solution of such an equation and the system is not synchronized, but for K large, non-trivial
solutions appear (synchronization). In the easiest instances, such a non-trivial solution is unique (in
the sense that r = ΨK,µ (r) admits a unique non-zero solution but of course one obtains an infinite
number of solutions by rotation invariance that is ψ can be chosen arbitrarily).
In [9], Dai Pra and den Hollander have rigorously shown the convergence of the averaged empirical
measure L N ∈ M1 (C ([0, T ], S1 ) × R) (probability measure on the whole trajectories and the
disorder):
N
1X
LN =
δ j,N
.
N j=1 (x ,ω j )
This convergence of the law of L N under the joint law of the oscillators and the disorder is shown
via an averaged large deviations principle in the case where b(x, y, ω) = K · f ( y − x) and
c(x, ω) = g(x, ω) for f and g smooth and bounded functions. As a corollary, it is deduced in
[9] the convergence of L N and of ν N , via a contraction principle, in the averaged set-up. In the
case of unbounded disorder, the same proof can be generalized (thesis in preparation) under the
following assumption:
Z
∀t > 0,
R
e t|ω| µ( dω) < ∞.
(HµA)
One aim of this paper is to obtain the limit of ν N ,(ω) in the quenched model, namely for a fixed
realization of the disorder (ω). This result can be deduced from the large deviations estimates in
[9], via a Borel-Cantelli
argument, but our result is more direct and works under the much weaker
R
assumption on µ, R |ω|µ( dω) < ∞.
A crucial aspect of the quenched convergence result, which is a law of large numbers, is that it shows
the self-averaging character of this limit: every typical disorder configuration leads as N → ∞ to the
same evolution equation.
However, it seems quite clear even at a superficial level that if we consider the central limit theorem
associated to this convergence, self-averaging does not hold since the fluctuations of the disorder
compete with the dynamical fluctuations. This leads for example to a remarkable phenomenon
(pointed out e.g. in [2] on the basis of numerical simulations): even if the distribution µ is symmetric, the fluctuations of a fixed chosen sample of the disorder makes it not symmetric
and thus
p
the center of the synchronization of the system slowly (i.e. with a speed of order 1/ N ) rotates in
one direction and with a speed that depends on the sample of the disorder (Fig. 1 and 2). This
non-self averaging phenomenon can be tackled in the sine-model by computing the finite-size order
parameters (Fig. 2):
N
D
E
j,N
1X
N ,(ω) iψNt ,(ω)
N ,(ω)
=
rt
e
ei x t = νt
, ei x .
(2)
N j=1
As a step toward understanding this non self-averaging phenomenon, the second and main goal of
this paper is to establish a fluctuations result around the McKean-Vlasov limit in the quenched set-up
(see Theorem 2.10). A central limit theorem for the averaged model is addressed in [9], applying
techniques introduced by Bolthausen [6]. A fluctuations theorem may also be found in [8] for a
794
Figure 1: We plot here the evolution of the marginal on S1 of ν N ,(ω) for N = 600 oscillators in
the sine-model (µ = 12 (δ−1 + δ1 ), K = 6). The oscillators are initially chosen independently and
uniformly on [0, 2π] independently of the disorder. First the dynamics leads to synchronization
of the oscillators (t = 6) to a profile which is close to a non-trivial stationary solution of McKeanN ,(ω)
Vlasov equation. We then observe that the center ψ t
of this density moves to the right with
an approximately constant speed; what is more, this speed of fluctuation turns out to be sampledependent (see Fig. 2).
795
Figure 2: Trajectories of the center of synchronization ψN ,(ω) in the sine-model for different realizations of the disorder (µ = 21 (δ−0.5 + δ0.5 ), K = 4, N = 400). We observe here the non self-averaging
phenomenon: direction and speed of the center depend on the choice of the initial
N -sample of the
p
disorder. Moreover, these simulations are compatible with speeds of order 1/ N .
796
model of social interaction in an averaged set-up. Here we prove convergence in law of the quenched
fluctuations process
Š
p €
ηN ,(ω) := N ν N ,(ω) − P ,
seen as a continuous process in the Schwartz space S ′ of tempered distributions on S1 × R to the
solution of an Ornstein-Uhlenbeck process. The quenched convergence is here understood as a weak
convergence in law w.r.t. the disorder and is more technically involved than the convergence in the
averaged system. The main techniques we exploit have been introduced by Fernandez and Méléard
[12] and Hitsuda and Mitoma [15], who studied similar fluctuations in the case without disorder.
In [7], A Large Deviation Principle is also proved. We refer to Section 4 for detailed definitions.
While numerical computations of the trajectories of the limit process of fluctuations clearly show a
non self-averaging phenomenon, the dynamical properties of the fluctuations process that we find
are not completely understood so far. Progress in this direction requires a good understanding
of the spectral properties of the linearized operator of McKean-Vlasov equation around its non1
has been treated
trivial stationary solution; the stability of the non-synchronized solution q ≡ 2π
by Strogatz and Mirollo in [24]. In the particular case without disorder, spectral properties of the
evolution operator linearized around the non-trivial stationary solution are obtained in [3], but the
case with a general distribution µ needs further investigations.
This work is organized as follows: Section 2 introduces the model and the main results. Section 3
focuses on the quenched convergence of νN . In Section 4, the quenched Central Limit Theorem is
proved. The last section 5 applies the fluctuations result to the behaviour of the order parameters
in the sine-model.
2 Notations and main results
2.1 Notations
• if X is a metric space, BX will be its Borel σ-field,
p
• C b (X ) (resp. C b (X ), p = 1, . . . , ∞), the set of bounded continuous functions (resp. bounded
continuous with bounded continuous derivatives up to order p) on X , (X will be often S1 ×R),
p
• Cc (X ) (resp. Cc (X ), p = 1, . . . , ∞), the set of continuous functions with compact support
(resp. continuous with compact support with continuous derivatives up to order p) on X ,
• D([0, T ], X ), the set of right-continuous with left limits functions with values on X , endowed
with the Skorokhod topology,
• M1 (Y ), the set of probability measures on Y (Y topological space, with a regular σ-field B),
• M F (Y ), the set of finite measures on Y ,
• (M1 (Y ), w): M1 (Y ) endowed with the topology of weak
R convergence, namely the coarsest topology on M1 (Y ) such that the evaluations ν 7→ f dν are continuous, where f are
bounded continuous,
797
• (M1 (Y ), v): M1 (Y ) endowed with the topology of
R vague convergence, namely the coarsest
topology on M1 (Y ) such that the evaluations ν 7→ f dν are continuous, where f are continuous with compact support.
We will use C as a constant which may change from a line to another.
2.2 The model
We consider the solutions of the following system of SDEs:
for i = 1, . . . , N , for T > 0, for all t ≤ T ,
i,N
xt
i
=ξ +
N
1X
N
j=1
Z
Z
t
t
b(x si,N , x sj,N , ω j ) ds +
c(x si,N , ωi ) ds + B ti ,
0
(3)
0
where the initial conditions ξi are independent and identically distributed with law λ, and independent of the Brownian motion (B) = (B i )i≥1 , and where b (resp. c) is a smooth function, 2π-periodic
w.r.t. the two first (resp. first) variables. The disorder (ω) = (ωi )i≥1 is a realization of i.i.d. random
variables with law µ.
Remark 2.1.
The assumption that the random variables (ωi ) are independent will not always be necessary and
will be weakened when possible.
Instead of considering x i,N as elements of R, we will consider their projection on S1 . For simplicity,
we will keep the same notation x i,N for this projection1 .
We introduce the empirical measure ν N (on the trajectories and disorder):
Definition 2.2.
For all t ≤ T , for a fixed trajectory (x 1 , . . . , x N ) ∈ C ([0, T ], (S1 )N ) and a fixed sequence of disorder
(ω), we define an element of M1 (S1 × R) by:
N ,(ω)
νt
=
N
1X
N
i=1
δ(x i,N ,ω ) .
t
i
Finally, we introduce the fluctuations process ηN ,(ω) of ν N ,(ω) around its limit P (see Th. 2.5):
Definition 2.3.
For all t ≤ T , for fixed (ω) ∈ RN , we define:
N ,(ω)
ηt
=
p N ,(ω)
N νt
− Pt .
Throughout this article, we will denote as P the law of the sequence of Brownian Motions and as P
the law of the sequence of the disorder. The corresponding expectations will be denoted as E and E
respectively.
1
See Remarks 2.7 and 2.12 for possible generalizations to the non-compact case.
798
2.3 Main results
2.3.1
Quenched convergence of the empirical measure
In [9], Dai Pra and den Hollander are interested in the averaged model, i.e. in the convergence in
law of the empirical measure under the joint law of both oscillators and disorder. The model studied
here, which is more interesting as far as the biological applications are concerned is quenched: for
a fixed realization of the disorder (ω), do we have the convergence of the empirical measure?
Moreover the convergence is shown under weaker assumptions on the moments of the disorder.
We consider here the general case where b(x, y, ω) is bounded, Lipschitz-continuous, and 2πperiodic w.r.t. the two first variables. c is assumed to be Lipschitz-continuous w.r.t. its first variable,
but not necessarily bounded (see the sine-model, where c(x, ω) = ω). We also suppose that the
function ω 7→ S(ω) := sup x∈S1 |c(x, ω)| is continuous (this is in particular true if c is uniformly
continuous w.r.t. to both variables (x, ω), and obvious in the sine-model where S(ω) = |ω|). The
Lipschitz bounds for b and c are supposed to be uniform in ω.
The disorder (ω), is assumed to be a sequence of identically distributed random variables (but not
necessarily independent), such that the law of each ωi is µ. We suppose that the sequence (ω)
satisfies the following property: for P-almost every sequence (ω),
N
1X
N
Z
sup |c(x, ωi )| →N →∞
1
i=1 x∈S
sup |c(x, ω)|µ( dω) < ∞.
x∈S1
(HµQ )
We make the following hypothesis on the initial empirical measure:
N ,(ω) N →∞
ν0
−→ ν0 ,
in law, in (M1 (S1 × R), w).
(H0 )
Remark 2.4. 1. The required hypotheses about the disorder and the initial conditions are weaker
than for the large deviation principle:
• the (identically distributed) variables (ωi ) need not be independent: we simply need a
convergence (similar to a law of large numbers) only concerning the function S,
• Condition (HµQ ) is weaker than (HµA) on page 796; for the sine-model, (HµQ ) reduces to
R
|ω|µ( dω) < ∞,
• the initial values need not be independent, we only assume a convergence of the empirical measure.
2. The hypothesis (HµQ ) is verified, for example, in the case of i.i.d. random variables, or in the
case of an ergodic stationary Markov process.
3. Under (H0 ), the second marginal of ν0 is µ.
In Section 3, we show the following:
Theorem 2.5.
Under the hypothesis (H0 ) and (HµQ ), for P-almost every sequence (ωi ), the random variable ν N ,(ω)
converges in law to P, in the space D([0, T ], (M1 (S1 × R), w)), where P is the only solution of the
799
following weak equation (for every f continuous bounded on S1 × R, twice differentiable, with bounded
derivatives):
Z t
Z t
1
′′ Pt , f = ν0 , f +
ds Ps , f +
ds Ps , f ′ (b[·, Ps ] + c) ,
(4)
2 0
0
where
Z
b[x, m] =
b(x, y, π)m( d y, dπ).
Moreover, with the same hypotheses, the law of ν N under the joint law of the oscillators and disorder
(averaged model) converges weakly to P as well.
Remark 2.6.
An easy calculation shows that P can be considered as a weak solution to the family of coupled
McKean-Vlasov equations (see [9]):
1. P can be written as P( dx, dω) = µ(ω)P ω ( dx),
ω
ω
2. if we define qω
t through Pt ( dx, dω) = µ(ω)q t ( dx), q t is the unique weak solution of the
McKean-Vlasov equation:
d
dt
ω ω
qω
t = L qt ,
q0ω = λ.
where, L ω is the following differential operator:
Œ ™
–‚Z
∂
1 ∂2 ω
ω ω
π
q .
L qt = −
b(x, y, π)q t ( d y)µ( dπ) + c(x, ω) qω
+
t
∂x
2 ∂ x2 t
R
(5)
(6)
We insist on the fact that Eq. (5) is indeed a (possibly) infinite system of coupled non-linear PDEs.
To fix ideas, one may consider the simple case where µ = 21 (δ−1 + δ1 ). Then (5) reduces to two
−1
equations (one for +1, the other for −1) which are coupled via the averaged measure 12 (q+1
t + q t ).
But for more general situations (µ = N (0, 1) say) this would consist of an infinite number of coupled
equations.
Remark 2.7 (Generalization to the non compact case).
The assumption that the state variables are in S1 , although motivated by the Kuramoto model, is not
absolutely essential: Theorem 2.5 still holds in the non-compact case (e.g. when S1 is replaced by
Rd ), under the additional
R assumptions of boundedness of x 7→ |c(x, ω)| and the first finite moment
of the initial condition: |x|λ( dx) < ∞.
We know turn to the statement of the main result of the paper: Theorem 2.10.
800
2.3.2
Quenched fluctuations of the empirical measure
Theorem 2.5 says that for P-almost every realization (ω) of the disorder, we have the convergence
of ν N ,(ω) towards P, which is a law of large numbers. We are now interested in the corresponding
Central Limit Theorem associated to this convergence, namely, for a fixed realization of the disorder
(ω), in the asymptotic behaviour, as N → ∞ of the fluctuations field ηN ,(ω) taking values in the set
of signed measures:
p N ,(ω)
N ,(ω)
∀t ∈ [0, T ], η t
:= N ν t
− Pt .
In the case with no disorder, such fluctuations have already been studied by numerous authors
(eg. Sznitman [25], Fernandez-Méléard [12], Hitsuda-Mitoma [15]). More particularly, Fernandez
and Méléard show the convergence of the fluctuations field in an appropriate Sobolev space to an
Ornstein-Uhlenbeck process.
Here, we are interested in the quenched fluctuations, in the sense that the fluctuations are studied
for fixed realizations of the disorder. We will prove a weak convergence of the law of the process
ηN ,(ω) , in law w.r.t. the disorder.
In addition to the hypothesis made in §2.3.1, we make the following assumptions about b and c
(where D p is the set of all differential operators of the form ∂uk ∂πl with k + l ≤ p):

b ∈ C b∞ (S1 × R), c ∈ C ∞ (S1 × R),
Z
supu∈S1 |Dc(u, π)|2

dπ < ∞,
∃α > 0, sup
1 + |π|2α
D∈D6 R

(H b,c )
Furthermore, we make the following assumption about the law of the disorder (α is defined in
(H b,c )):
Z
the (ω j ) are i.i.d. and
R
|ω|4α µ( dω) < ∞.
(HµF )
Remark 2.8.
The regularity hypothesis about b and c can be weakened (namely b ∈ C bn (S1 ×R) and c ∈ C m (S1 ×
R) for sufficiently large n and m) but we have kept m = n = ∞ for the sake of clarity.
Remark 2.9.
In the case of the sine-model, Hypothesis (H b,c ) is satisfied with α = 2 for example.
In order to state the fluctuations theorem, we need some further notations: for all s ≤ T , let Ls be
the second order differential operator defined by
Ls (ϕ)( y, π) :=
1
2
ϕ ′′ ( y, π) + ϕ ′ ( y, π)(b[ y, Ps ] + c( y, π)) + Ps , ϕ ′ (·, ·)b(·, y, π) .
Let W the Gaussian process with covariance:
Z
s∧t
E(Wt (ϕ1 )Ws (ϕ2 )) =
0
801
¬
¶
Pu , ϕ1′ ϕ2′ du.
(7)
For all ϕ1 , ϕ2 bounded and continuous on S1 × R, let
Z
Covλ ϕ1 (· , ω), ϕ2 (· , ω) µ( dω),
Γ1 (ϕ1 , ϕ2 ) =
R
Z
‚
=
ϕ1 −
S1 ×R
Œ‚
Z
ϕ1 (· , ω) dλ
(8)
ϕ2 −
S1
Œ
Z
ϕ2 (· , ω) dλ λ( dx)µ( dω),
S1
and
‚Z
Γ2 (ϕ1 , ϕ2 ) = Covµ
ϕ2 dλ ,
ϕ1 dλ,
S1
S1
Z ‚Z
S1
ϕ1 dλ −
(9)
Œ ‚Z
Z
=
R
Œ
Z
ϕ1 dλ dµ
S1
S1 ×R
Œ
Z
ϕ2 dλ −
ϕ2 dλ dµ
dµ.
S1 ×R
For fixed (ω), we may consider HN (ω), the law of the process ηN ,(ω) ; HN (ω) belongs to
M1 (C ([0, T ], S ′ )), where S ′ is the usual Schwartz space of tempered distributions on S1 × R.
We are here interested in the law of the random variable (ω) 7→ HN (ω) which is hence an element
of M1 (M1 (C ([0, T ], S ′ ))).
The main theorem (which is proved in Section 4) is the following:
Theorem 2.10 (Fluctuations in the quenched model).
Under (HµF ), (H b,c ), the sequence (ω) 7→ HN (ω) converges in law to the random variable ω 7→ H (ω),
where H (ω) is the law of the solution to the Ornstein-Uhlenbeck process ηω solution in S ′ of the
following equation:
Z
t
ηω
t = X (ω) +
0
Ls∗ ηω
s ds + Wt ,
(10)
where, Ls∗ is the formal adjoint operator of Ls and for all fixed ω, X (ω) is a non-centered Gaussian
process with covariance Γ1 and with mean value C(ω). As a random variable in ω, ω 7→ C(ω) is a
Gaussian process with covariance Γ2 . Moreover, W is independent on the initial value X .
Remark 2.11.
In the evolution (10), the linear operator Ls∗ is deterministic ; the only dependence in ω lies in
the initial condition X (ω), through its non trivial means C(ω). However, numerical simulations of
trajectories of ηω (see Fig. 3) clearly show a non self-averaging phenomenon, analogous to the one
observed in Fig 2: ηω
t not only depends on ω through its initial condition X (ω), but also for all
positive time t > 0.
Understanding how the deterministic operator Ls∗ propagates the initial dependence in ω on the
whole trajectory is an intriguing question which requires further investigations (work in progress).
In that sense, one would like to have a precise understanding of the spectral properties of Ls∗ , which
appears to be deeply linked to the differential operator in McKean-Vlasov equation (6) linearized
around its non-trivial stationary solution.
Remark 2.12 (Generalization to the non-compact case).
As in Remark 2.7, it is possible to extend Theorem 2.10 to the (analogous but more technical) case
802
where S1 is replaced by Rd . To this purpose, one has to introduce an additional weight (1+|x|α )−1 in
the definition of the Sobolev norms in Section
R 4 and to suppose appropriate hypothesis concerning
the first moments of the initial condition λ ( |x|β λ( dx) < ∞ for a sufficiently large β).
2.3.3
Fluctuations of the order parameters in the Kuramoto model
N ,(ω)
For given N ≥ 1, t ∈ [0, T ] and disorder (ω) ∈ RN , let r t
N ,(ω) N ,(ω)
ζt
rt
=
N
1X
N
N ,(ω)
> 0 and ζ t
∈ S1 such that
D
E
j,N
N ,(ω)
ei x t = νt
, ei x .
j=1
N ,(ω)
Proposition 2.13 (Convergence and fluctuations for r t
We have the following:
).
N ,(ω)
1. Convergence of r t
: For P-almost every realization of the disorder, r N ,(ω) converges in law in
C ([0, T ], R), to r defined by
t ∈ [0, T ] 7→ r t :=
€
Pt , cos(·)
2
2 Š 21
.
+ Pt , sin(·)
2. If r0 > 0 then
∀t ∈ [0, T ],
N ,(ω)
3. Fluctuations of r t
r t > 0.
(H r )
around its limit: Let
N ,(ω)
t 7→ R t
:=
p N ,(ω)
N rt
− rt
be the fluctuations process.
For fixed disorder (ω), let RN ,(ω) ∈ M1 (C ([0, T ], R))
N ,(ω)
be the law of R
. Then, under (H r ), the random variable (ω) 7→ RN ,(ω) converges in law to the random variable ω 7→ Rω , where Rω is the law of R ω :=
1
⟨P , cos(·)⟩ · ηω , cos(·) + ⟨P , sin(·)⟩ · ηω , sin(·) .
r
Remark 2.14.
In simpler terms, this double convergence in law corresponds for example to the convergence in
law of the corresponding characteristic functions (since the tightness is a direct consequence of
the tightness of the process η); i.e. for t 1 , . . . , t p ∈ [0, T ] (p ≥ 1) the characteristic function of
N ,(ω)
N ,(ω)
, . . . , R t p ) for fixed (ω) converges in law, as a random variable in (ω), to the random
(R t 1
characteristic function of (R tω , . . . , R tω ).
1
p
Proposition 2.15 (Convergence and fluctuations for ζN ,(ω) ).
We have the following:
1. Convergence of ζN ,(ω) : Under (H r ), for P-almost every realization of the disorder (ω), ζN ,(ω)
⟨P , ei x ⟩
converges in law to ζ : t ∈ [0, T ] 7→ ζ t := t r
,
t
803
Figure 3: We plot here the evolution of the imaginary part of the process Z ω , for different realizations of ω; the trajectories are sample-dependent, as in Fig 2.
2. Fluctuations of ζN ,(ω) around its limit: Let
N ,(ω)
t 7→ Z t
:=
p N ,(ω)
N ζt
− ζt
be the fluctuations process. For fixed disorder (ω), let ZN ,(ω) ∈ M1 (C ([0, T ], R)) be the law of
Z N ,(ω) . Then, under (H r ), the random variable (ω)
ZN ,(ω) converges
in law
€ 7→
¶ toŠthe random
ω
¬
1
ω
ω
ω
variable ω 7→ Z , where Z is the law of Z := r 2 r η , cos(·) + P , e i x R ω .
N ,(ω)
In the sine-model, we have ζN ,(ω) = e iψ
where ψN ,(ω) is defined in (2) and is plotted in Fig. 2.
ω
Some trajectories of the process Z are plotted in Fig. 3.
This fluctuations result is proved in Section 5.
3 Proof of the quenched convergence result
In this section we prove Theorem 2.5. Reformulating (3) in terms of ν N ,(ω) , we have:
Z
∀i = 1 . . . N , ∀t ∈ [0, T ],
where we recall that b[x, m] :=
R
i,N
xt
Z
t
i
t
b[x si,N , νsN ] ds +
=ξ +
0
c(x si,N , ωi ) ds + B ti ,
(11)
0
b(x, y, π)m( d y, dπ).
The idea of the proof of Theorem 2.5 is the following: we show the tightness of the sequence (ν N ,(ω) )
firstly in D([0, T ], (M F , v)) (recall Notations in §2.1), which is quite simple since Cc (S1 × R) is
804
separable and by an argument of boundedness of the second marginal of any accumulation point,
thanks to (HµQ ), we show the tightness in D([0, T ], (M F , w)). The proof is complete when we prove
the uniqueness of any accumulation point.
3.1 Proof of the tightness result
We want to show successively:
1. Tightness of L (ν N ,(ω) ) in D([0, T ], (M F , v)),
2. Equation verified by any accumulation point,
3. Characterization of the marginals of any limit,
4. Convergence in D([0, T ], (M F , w)).
3.1.1
Equation verified by ν N ,(ω)
For f ∈ C b2 (S1 × R), we denote by f ′ , f ′′ the first and second derivative of f with respect to the first
R
variable. Moreover, if m ∈ M1 (S1 × R), then m , f stands for S1 ×R f (x, π)m( dx, dπ).
Applying Ito’s formula to (11), we get, for all f ∈ C b2 (S1 × R),
D
N ,(ω)
νt
,
E
f
D
=
N ,(ω)
ν0
,
Z
t
+
0
where MN , f (t) :=
3.1.2
1
N
PN R t
j=1
0
f
+
1
Z
2
t
¬
¶
ds νsN ,(ω) , f ′′
0
¬
¶
ds νsN ,(ω) , f ′ · (b[·, νsN ,(ω) ] + c) + MN , f (t),
N ,(ω)
f ′ (x j
E
, ω j ) dB j (s) is a martingale ( f ′ bounded).
Tightness of L (ν N ,(ω) ) in D([0, T ], (M F , v))
Cc (S1 × R) is separable: let ( f k )k≥1 (elements of C ∞ (S1 × R)) a dense sequence in Cc (S1 × R), and
let f0 ≡ 1. We define Ω := D([0, T ], (M1 , v)) and the applications Π f , f ∈ Cc (S1 × R) by:
Πf :
Ω → D([0, T ], R)
m 7→
m, f .
Let (Pn )n a sequence of probabilities on Ω and (Π f Pn ) = Pn ◦ Π−1
∈ D([0, T ], R). We recall the
f
following result:
Lemma 3.1.
If for all k ≥ 0, the sequence (Π f k Pn )n is tight in M1 (D([0, T ], R)), then the sequence (Pn )n is tight in
M1 (D([0, T ], (M1 , v))).
805
Hence, it suffices to have a criterion for tightness in D([0, T ], R). Let X tn be a sequence of
processes in D([0, T ], R) and F tn a sequence of filtrations such that X n is F n -adapted. Let
φ n = {stopping times for F n }. We have (cf. Billingsley [4]):
Lemma 3.2 (Aldous’ criterion).
If the following holds,
€
Š
1. L sup t≤T X tn n is tight,
2. For all ǫ > 0 and η > 0, there exists δ > 0 such that
Š
€
lim sup
sup
P X Sn − X Sn′ > η ≤ ǫ,
n
S,S ′ ∈φ n ;S≤S ′ ≤(S+δ)∧T
then L (X n ) is tight.
Proposition 3.3.
The sequence L (ν N ,(ω) ) is tight in D([0, T ], (M F , v)).
Proof. For all ǫ > 0, for all k ≥ 1 (the case k = 0 is straightforward),


‚
Œ
D
E
D
E 

N ,(ω)
1
N
,(ω)
P sup ν t
, fk >
, 1 
≤ ǫ f k ∞ E 
sup ν t
,

ǫ
t≤T
t≤T |
{z
}
(Markov Inequality).
=1
The tightness follows.
For all k ≥ 1, we have the following decomposition:
D
E D
E
N ,(ω)
N ,(ω)
N ,(ω)
N ,(ω)
νt
, f k = ν0
, fk + At
( fk ) + Mt
( f k ),
N ,(ω)
N ,(ω)
where A t
( f k ) is a process of bounded variations, and M t
( f k ) is a square-integrable martingale. Then it suffices to verify Lemma 3.2, (2) for A and M separately. For all ǫ > 0 and η > 0, for
all stopping times S, S ′ ∈ φ N ; S ≤ S ′ ≤ (S + δ) ∧ T , we have:
‹

N ,(ω)
N ,(ω)
( f k ) > η ,
aN := P AS ′ ( f k ) − AS


Z ′
 Z ′
S
S
¬
¬
¶
¶
1
1
1
≤ E
ds νsN ,(ω) , f k′ · (b[·, νsN ,(ω) ] + c)  + E 
ds νsN ,(ω) , f k′′  ,
η
η
2
S
S
≤
C ′
E S − S ≤ ǫ,
η
for δ sufficiently small.
(we use here that f k are of compact support for k ≥ 1; in particular the function (x, π) 7→
f k′ (x, π)c(x, π) is bounded). Furthermore,
‹

2
N ,(ω)
N ,(ω)
N ,(ω)
N ,(ω)
2
( f k ) − MS
( f k ) > η ,
( f k ) − MS
( f k ) > η = P MS ′
P MS ′
2 1
N ,(ω)
N ,(ω)
≤ 2 E MS ′
( f k ) − MS
( f k ) ,
η


N Z S′
X
C
1
f k′2 (x si , ωi ) ds ≤
E
δ.
≤
2
(N η)
N η2
i=1 S
806
At this point, L (ν N ,(ω) ) is tight in D([0, T ], (M F , v)).
3.1.3
Equation satisfied by any accumulation point in D([0, T ], (M F , v))
Using hypothesis (H0 ), it is easy to show that the following equation is satisfied for every accumulation point ν, for every f ∈ Cc2 (S1 × R) (we use here that S1 is compact):
Z
t
ν t , f = ν0 , f +
0
′
ds νs , f · (b[·, νs ] + c) +
1
Z
2
t
ds νs , f ′′ .
(12)
0
For any accumulation point ν, the following lemma gives a uniform bound for the second marginal
of ν:
Lemma 3.4.
Let Q be an accumulation point of L (ν N ,(ω) )N in D([0, T ], (M1 , v)) and let be ν ∼ Q. For all t ∈
[0, T ], we define by (ν t,2 ) the second marginal of ν t :
Z
∀A ∈ B(R),
(ν t,2 )(A) =
ν t ( dx, dπ).
S1 ×A
Then, for all t ∈ [0, T ],
Z
Z
sup |c(x, π)| µ( dπ).
sup |c(x, π)| (ν t,2 )( dπ) ≤
R x∈S1
R x∈S1
Proof. Let φ be a C 2 positive function such that φ ≡ 1 on [−1, 1], φ ≡ 0 on [−2, 2] and φ ∞ ≤ 1.
Let,
π‹
∀k ≥ 1, φk := π 7→ φ
.
k
φ ,
1,
for
all
π.
We
have
also
for
all
π
∈
R,
|φ
(π)|
≤
Then φk ∈ Cc2(R) and φk (π)→k→∞
k
∞
′′ ′′
′
′
|φk (π)| ≤ φ ∞ , |φk (π)| ≤ φ ∞ .
We have successively, denoting S(π) := sup x∈S1 |c(x, π)|,
Z
Z
S(π)ν t ( dx, dπ) =
S1 ×R
lim inf φk (π)S(π)ν t ( dx, dπ),
S1 ×R
≤ lim inf
k→∞
k→∞
Z
φk (π)S(π)ν t ( dx, dπ), (Fatou’s lemma),
S1 ×R
Z
N ,(ω)
= lim inf lim
k→∞ N →∞
Z
≤ lim
N →∞
φk (π)S(π)ν t
( dx, dπ),
(13)
S1 ×R
N ,(ω)
S(π)ν t
S1 ×R
( dx, dπ), (since φ ∞ ≤ 1).
The equality (13) is true since (x, π) 7→ φk (π)S(π) is of compact support in S1 × R (recall that S is
supposed to be continuous by hypothesis).
807
N ,(ω)
But, by definition of ν t
, and using the hypothesis (HµQ ) concerning µ, we have,
Z
Z
N ,(ω)
S(π)ν t
( dx,
lim
N →∞
S1 ×R
dπ) =
S(π)µ( dπ).
(14)
R
The result follows.
Remark 3.5.
(14) is only true for P-almost every sequence (ω). We assume that the sequence (ω) given at the
beginning satisfies this property.
3.1.4
Tightness in D([0, T ], (M F , w))
We have the following lemma (cf. [21]):
Lemma 3.6.
Let (X n ) be a sequence of processes in D([0, T ], (M F , w)) and X a process belonging to
C ([0, T ], (M F , w)). Then,
(
L
Xn → X
in D([0, T ], (M F , v)),
L
Xn → X ⇔
L
X n , 1 → ⟨X , 1⟩ in D([0, T ], R).
So, it suffices to show, for any accumulation point ν:
1. ⟨ν , 1⟩ = 1: Eq. (12) is true for all f ∈ Cc2 (S1 × R), so in particular for f k (x, π) := φk (π).
Using the boundedness shown in lemma 3.4, we can apply dominated convergence theorem
in Eq. (12). We then have ν t , 1 = 1, for all t ∈ [0, T ]. The fact that Eq. (12) is verified for
all f ∈ C b2 (S1 × R) can be shown in the same way.
2. Continuity of the limit: For all 0 ≤ s ≤ t ≤ T , for all f ∈ C b2 (S1 × R),
ν , f − ν , f ≤ K
t
s
Z
Z
s
t
+
s
t
1
ν , f ′ · b[·, ν ] du +
u
u
2
Z
ν , f ′ · c du ≤ C × |t − s| ,
u
t
ν , f ′′ du
u
s
for some constant C.
Noticing that we used again Lemma 3.4 to bound the last term, we have the result.
Remark 3.7.
It is then easy to see that the second marginal (on the disorder) of any accumulation point is µ.
At this point L (ν N ,(ω) ) is tight in D([0, T ], (M F , w)). It remains to show the uniqueness of any
accumulation point: it shows firstly that the sequence effectively converges and that the limit does
not depend on the given sequence (ω).
808
3.2 Uniqueness of the limit
Proposition 3.8.
There exists a unique element P of D([0, T ], M F (S1 × R)) which satisfies equation (12), P0 ∈ M1 and
P0,2 = µ.
Remark 3.9.
Since this proof is an adaptation of Oelschläger [20] Lemma 10, p.474, we only sketch the proof:
We can rewrite Eq. (12) in a more compact way:
Z
t
Pt , f = P0 , f +
Ps , L(Ps )( f ) ds,
0
where,
L(P)( f )(x, ω) :=
1
2
∀ f ∈ C b2 (S1 × R), 0 ≤ t ≤ T,
f ′′ (x, ω) + h(x, ω, P) f ′ (x, ω),
(15)
∀x ∈ S1 , ∀ω ∈ R,
and,
h(x, π, P) = b[x, P] + c(x, π).
Let t 7→ Pt be any solution of Eq. (15). One can then introduce the following SDE (where ξ ∈ R and
ξ̃ ∈ S1 is its projection on S1 ):
¨
dξ t
dω t
= h(ξ̃ t , ω t , Pt ) dt + dWt ,
= 0.
ξ0 = ξ̃0 , (ξ̃0 , ω0 ) ∼ P0 ,
(16)
Eq. (16) has a unique (strong) solution (ξ t , ω t ) t∈[0,T ] = (ξ t , ω0 ) t∈[0,T ] . The proof of uniqueness in
Eq. (12) consists in two steps:
1. Pt = L (ξ̃ t , ω t ), for all t ∈ [0, T ],
2. Uniqueness of ξ̃.
We refer to Oelschläger [20] for the details. Another proof of uniqueness can be found in [9] or in
[13] via a martingale argument.
4 Proof of the fluctuations result
Now we turn to the proof of Theorem 2.10. To that purpose, we need to introduce some distribution
spaces:
809
4.1 Distribution spaces
Let S := S (S1 × R) be the usual Schwartz space of rapidly decreasing infinitely differentiable
functions. Let D p be the set of all differential operators of the form ∂uk ∂πl with k + l ≤ p. We know
from Gelfand and Vilenkin [14] p. 82-84, that we can introduce on S a nuclear Fréchet topology
by the system of seminorms k · k p , p = 1, 2, . . . , defined by
p Z
X
2 X
φ =
|Dφ( y, π)|2 d y dπ.
(1 + |π|2 )2p
p
S1 ×R
k=0
D∈Dk
Let S ′ be the corresponding dual space of tempered distributions. Although, for the sake of simplicity, we will mainly consider ηN ,(ω) as a process in C ([0, T ], S ′ ), we need some more precise
estimations to prove tightness and convergence. We need here the following norms:
For every integer j, α ∈ R+ , we consider the space of all real functions ϕ defined on S1 × R with
derivative up to order j such that

1/2
Z
2
X
|∂ x k1 ∂πk2 ϕ(x, π)|

ϕ := 
dx dπ < ∞.

j,α
2α
1 + |π|
k +k ≤ j S1 ×R
1
2
j,α
− j,α
j,α
Let W0 be the completion of Cc∞ (S1 × R) for this norm; (W0 , k · k j,α ) is a Hilbert space. Let W0
be its dual space.
Let C j,α be the space of functions ϕ with continuous partial derivatives up to order j such that
|∂ x k1 ∂πk2 ϕ(x, π)|
lim sup
1 + |π|α
|π|→∞ x∈S1
= 0, for all k1 + k2 ≤ j,
with norm
ϕ
C j,α
X
=
sup sup
|∂ x k1 ∂πk2 ϕ(x, π)|
1
k1 +k2 ≤ j x∈S π∈R
1 + |π|α
.
We have the following embeddings:
m+ j,α
W0
,→ C j,α , m > 1, j ≥ 0, α ≥ 0,
i.e. there exists some constant C such that
ϕ j,α ≤ C ϕ .
C
m+ j,α
Moreover,
m+ j,α
W0
j,α+β
,→ W0
Thus there exists some constant C such that
ϕ
(17)
, m > 1, j ≥ 0, α ≥ 0, β > 1.
j,α+β
≤ C ϕ m+ j,α .
We then have the following dual continuous embedding:
− j,α+β
W0
j,α
It is quite clear that S ,→ W0
−(m+ j),α
,→ W0
, m > 1, α ≥ 0, β > 1.
for any j and α, with a continuous injection.
We now prove some continuity of linear mappings in the corresponding spaces:
810
(18)
Lemma 4.1.
3,α
For every x, y ∈ S1 , ω ∈ R, for all α, the linear mappings W0 → R defined by
D x, y,ω (ϕ) := ϕ(x, ω) − ϕ( y, ω); D x,ω := ϕ(x, ω); H x,ω = ϕ ′ (x, ω),
are continuous and
D
x, y,ω
≤ C|x − y| (1 + |ω|α ) ,
−3,α
α
D x,ω −3,α ≤ C (1 + |ω| ) ,
α
H x,ω −3,α ≤ C (1 + |ω| ) .
(19)
(20)
(21)
Proof. Let ϕ be a function of class C ∞ with compact support on S1 × R, then,
|ϕ(x, ω) − ϕ( y, ω)| ≤ |x − y| sup ϕ ′ (u, ω) ,
u
!
ϕ ′ (u, ω)
≤ |x − y| (1 + |ω|α ) sup
,
1 + |ω|α
u,ω
≤ |x − y| (1 + |ω|α ) ϕ 1,α ,
≤ C|x − y| (1 + |ω|α ) ϕ 3,α ,
following (17) with j = 1 and m = 2 > 1. Then, (19) follows from a density argument. (20) and
(21) are proved in the same way.
4.2 The non-linear process
The proof of convergence is based on the existence of the non-linear process associated to McKeanVlasov equation. Such existence has been studied by numerous authors (eg. Dawson [10], JourdainMéléard [16], Malrieu [18], Shiga-Tanaka [23], Sznitman [25], [26]) mostly in order to prove some
propagation of chaos properties in systems without disorder. We consider the present similar case
where disorder is present. Let us give some intuition of this process. One can replace the nonlinearity in Eq. (5) by an arbitrary measure m( dx, dω):
∂ t qω
t =
1
2
”
—
ω
∂ x x qω
.
t − ∂ x (b[x, m] + c(x, ω)) q t
In this particular case, it is usual to interpret qω
t as the time marginals of the following diffusion:
ω
ω
d xω
t = d B t + b[x t , m]d t + c(x t , ω)d t, ω ∼ µ.
It is then natural to consider the following problem, where m is replaced by the proper measure P:
on a filtered probability space (Ω, F , F t , B, x 0 , Q), endowed with a Brownian motion B and with a
F0 measurable random variable x 0 , we introduce the following system:

Rt
Rt
ω
ω
ω
 x t = x 0 + 0 b[x s , Ps ] ds + 0 c(x s , ω) ds + B t ,
(22)
 ω ∼ µ,
Pt = L (x t , ω), ∀t ∈ [0, T ].
811
Proposition 4.2.
There is pathwise existence and uniqueness for Equation (22).
Proof. The proof is the same as given in Sznitman [26], Th 1.1, p.172, up to minor modifications.
The main idea consists in using a Picard iteration in the space of probabilities on C ([0, T ], S1 × R)
endowed with an appropriate Wasserstein metric. We refer to it for details.
4.3 Fluctuations in the quenched model
The key argument of the proof is to explicit the speed of convergence as N → ∞ for the rotators to
the non-linear process (see Prop. 4.3).
A major difference between this work and [12] is that, since in our quenched model, we only
integrate w.r.t. oscillators and not w.r.t. the disorder, one has to deal with remaining terms, see ZN
in Proposition 4.3, to compare with [12], Lemma 3.2, that would have disappeared in the averaged
model. The main technical difficulty of Proposition 4.3 is to control the asymptotic behaviour of such
terms, see (24). As in [12], having proved Prop. 4.3, the key argument of the proof is a uniform
estimation of the norm of the process ηN ,(ω) , see Propositions 4.4 and 4.8, based on the generalized
stochastic differential equation verified by ηN (ω) , see (30).
4.3.1
Preliminary results
We consider here a fixed realization of the disorder (ω) = (ω1 , ω2 , . . . ). On a common filtered
probability space (Ω, F , F t , (B i )i≥1 , Q), endowed with a sequence of i.i.d. F t -adapted Brownian
motions (Bi ) and with a sequence of i.i.d. F0 measurable random variables (ξi ) with law λ, we
define as x i,N the solution of (11), and as x ωi the solution of (22), with the same Brownian motion
B i and with the same initial value ξi .
The main technical proposition, from which every norm estimation of ηN ,(ω) follows is the following:
Proposition 4.3.
2 ™
i,N
ωi E sup x t − x t ≤ C/N + ZN (ω1 , . . . , ωN ),
–
(23)
t≤T
where the random variable (ω) 7→ ZN (ω) is such that:
lim lim sup P N ZN (ω) > A = 0.
A→∞ N →∞
(24)
The (rather technical) proof of Proposition 4.3 is postponed to the end of the document (see §A).
Once again, we stress the fact that the term ZN would have disappeared in the averaged model.
The first norm estimation of the process ηN ,(ω) (which will be used to prove tightness) is a direct
consequence of Proposition 4.3 and of a Hilbertian argument:
Proposition 4.4.
Under the hypothesis (HµF ) on µ, the process ηN ,(ω) satisfies the following property: for all T > 0,
N ,(ω) 2
≤ AN (ω1 , . . . , ωN ),
(25)
sup E η t
t≤T
−3,2α
812
where
lim lim sup P AN > A = 0.
A→∞ N →∞
3,2α
Proof. For all ϕ ∈ W0
D
N ,(ω)
ηt
, writing
N ¦
N ¦
E
©
1 X
1 X
©
ω
ω
i,N
,ϕ =p
ϕ(x t , ωi ) − ϕ(x t i , ωi ) + p
ϕ(x t i , ωi ) − Ps , ϕ ,
N i=1
N i=1
N ,(ω)
=: S t
we have:
N ,(ω)
(ϕ) + Tt
D
N ,(ω)
ηt
,ϕ
E2
(ϕ),
N ,(ω)
N ,(ω)
≤ 2 St
(ϕ)2 + Tt
(ϕ)2 .
(26)
But, by convexity,
N ,(ω)
St
(ϕ)2
≤
N
X
D2 i,N
i=1
ω
x t ,x t i ,ωi
(ϕ).
3,2α
Then, applying the latter equation to an orthonormal system (ϕ p ) p≥1 in the Hilbert space W0
summing on p, we have by Parseval’s identity on the continuous functional D x i,N ,x ωi ,
t
N ,(ω) 2
E St

−3,2α
≤ E
≤C
≤C
N 2
X
D x i,N ,x ωi ,ω t
i=1
N
X
i=1
−3,2α
t

,
2 i,N
ω (1 + |ωi |4α )E x t − x t i ,
N €
X
i=1
i
t
1 + |ωi |4α
Š
C/N + ZN (ω1 , . . . , ωN ) ,
where we used (19) in (27), and (23) in (28).
On the other hand,
h
N ,(ω)
E Tt
(ϕ)2
i
)2 
N
X
1 

ω
= E
(ϕ(x t i , ωi ) − Pt , ϕ )  ,
N
i=1


N
X
1
ω
(ϕ(x t i , ωi ) − Pt , ϕ )2 
= E
N
i=1


1 X

ωj
ω
+ E  (ϕ(x t i , ωi ) − Pt , ϕ )(ϕ(x t , ω j ) − Pt , ϕ ) ,
N
i6= j


N
X
2
1X
ω
2
≤ E
G(ϕ)(ωi )G(ϕ)(ω j ),
(ϕ(x t i , ωi )2 + Pt , ϕ ) +
N
N i6= j
i=1


!2
N
N
X
1 X
2
2
ω
2
i
ϕ(x t , ωi )  + 2 Pt , ϕ + p
G(ϕ)(ωi ) ,
≤ E
N
N i=1
i=1
(
813
,
(27)
(28)
where G(ϕ)(ω) :=
S N ,(ω) , we see
R
ω
ϕ( y, ωi )Pt i ( d y) − Pt , ϕ . If we apply the same Hilbertian argument as for
N ,(ω) 2
E Tt
−3,2α
4α 

≤
E
(1 + |ωi | ) + C + φ 7→
N
i=1
2C


N
X
N
1 X
G(φ)(ωi )
p
N i=1
! 2
,
(29)
−3,2α
It is easy to see that the last term in (29) can be reformulated as BN (ω1 , . . . , ωN ), with the property
that limA→∞ lim supN →∞ P(BN > A) = 0. Combining (24), (26), (28) and (29), Proposition 4.4 is
proved.
4.3.2
Tightness of the fluctuations process
Applying Ito’s formula to (11), we obtain, for all ϕ bounded function on S1 × R, with two bounded
derivatives w.r.t. x, for every sequence (ω), for all t ≤ T :
D
E D
E Z tD
E
N ,(ω)
ηt
N ,(ω)
, ϕ = η0
N
N ,(ω)
ηsN ,(ω) , Lsν (ϕ) ds + M t
,ϕ +
0
(ϕ),
(30)
where, for all y ∈ S1 , π ∈ R,
N
Lsν (ϕ)( y, π) =
N ,(ω)
and M t
1
2
€
Š ϕ ′′ ( y, π) + ϕ ′ ( y, π) b[ y, νsN ] + c( y, π) + Ps , ϕ ′ (·, ·)b(·, y, π) ,
(ϕ) is a real continuous martingale with quadratic variation process
¬
M
N ,(ω)
(ϕ)
Z
¶
t
t
¬
=
¶
νsN ,(ω) , ϕ ′ ( y, π)2 ds.
0
Lemma 4.5.
N
For every N , the operator Lsν defines a linear mapping from S into S and for all ϕ ∈ S ,
2
νN
Ls (ϕ) 3,2α
Proof. The terms
remaining terms:
1 ′′
ϕ ( y, π)
2
2
≤ C ϕ 6,α .
and ϕ ′ ( y, π)b[ y, νsN ] clearly satisfy the lemma. We study the two
X
P , ϕ ′ b(·, y, π) 2 =
s
3,2α
k1 +k2 ≤3
Z
¬
Z
Ps , ϕ ′ ∂ y k1 ∂πk2 b(·, y, π)
S1 ×R
1
Z
1 + |π|4α
¶2
d y dπ,
dπ
ϕ ′ ( y, π)2 Ps ( d y, dπ),
4α
1
+
|π|
S1 ×R
R
Z
Z
2
1
dπ
(1 + |π|α )2 Ps ( d y, dπ),
≤ C ϕ C 3,α
4α
1
+
|π|
1
ZR
Z S ×R
2
1
≤ C ϕ 6,α
dπ (1 + |π|α )2 µ( dπ).
4α
1
+
|π|
R
R
≤C
814
And,
ϕ ′ ( y, π)c( y, π) 2
3,2α
=
€
Z
X
k1 +k2 ≤3
Š 2
∂ y k1 ∂πk2 ϕ ′ ( y, π)c( y, π)
1 + |π|4α
S1 ×R
d y dπ.
It suffices to estimate, for every differential operator Di = ∂ y ui ∂πvi , i = 1, 2 with u1 +u2 + v1 + v2 ≤ 3,
the following term:
Z
Z
|D1 ϕ ′ ( y, π)D2 c( y, π)|2
|D1 ϕ ′ ( y, π)|2 |D2 c( y, π)|2 (1 + |π|α )2
d
y
dπ
≤
d y dπ,
α 2
1 + |π|4α
1 + |π|4α
S1 ×R
S1 ×R (1 + |π| )
Z
2
sup y∈S1 |D2 c( y, π)|2
dπ.
≤ C ϕ 6,α
1 + |π|2α
R
The result follows from the assumptions made on c.
For the tightness criterion used below, we need to ensure that the trajectories of the fluctuations
process are almost surely continuous: in that purpose, we need some more precise evaluations than
in Prop. 4.4.
Proposition 4.6.
N ,(ω)
The process (M t
) satisfies, for every (ω), and for every T > 0,
™
–
N €
Š
CX
N ,(ω) 2
≤
E sup M t
1 + |ωi |4α .
−3,2α
N i=1
t≤T
Remark 4.7.
In particular, a consequence of (HµF ) is that, for P-almost every sequence (ω),
™
–
N ,(ω) 2
sup E sup M t
N
t≤T
≤ sup
−3,2α
N
N €
CX
N
i=1
3,2α
Proof. Let (ϕ p ) p≥1 a complete orthonormal system in W0
h
i
P
N ,(ω)
2
E
sup
(M
(ϕ
))
is bounded by
p
t≤T
t
p≥1
Z
i
X h
X
N ,(ω)
C
E MT
(ϕ p )2 = C
E
p≥1
=
=
≤
N
N
C
N
Z
0
Z
νsN ,(ω) , ϕ p′ ( y, π)2
E
ds ,
N €
X
0

X
ϕ p′ (x si,N , ωi )2 ds,
p≥1
T
E
i=1
i=1
T
E
i=1
N
1X

D
2
H x si,N ,ω i
3,2α

ds,
Š
1 + |ωi |4α , (using (21)).
815
(31)
. For fixed N , by Doob’s inequality,
0
p≥1
N
1X
T
Š
1 + |ωi |4α < ∞.
Proposition 4.8.
For every N , every (ω),
™
–
N ,(ω) 2
E sup η t
−6,α
t≤T
with
< CN (ω1 , . . . , ωN ),
(32)
lim lim sup P CN > A = 0.
A→∞ N →∞
6,α
Proof. Let (ψ p ) be a complete orthonormal system in W0 of C ∞ function on S1 × R with compact
support. We prove the stronger result:


D
E2
X
N
,(ω)
E
, ψ p  < ∞.
sup η t
p≥1 t≤T
Indeed,
D
N ,(ω)
ηt
,
ψp
E2
‚
≤C
D
N ,(ω)
η0
,
ψp
E2
Z tD
+T
ηsN ,(ω) ,
0
By Doob’s inequality,


 E2
D
X
N ,(ω) 2
N
,(ω)
E
, ψ p  ≤ C E η0
sup η t
Z
T
+E
−6,α
p≥1 t≤T
N
Lsν (ψ p )
0
E2
XD
p≥1
Œ
ds +
N ,(ω)
Mt
(ψ p )2
N
ηsN ,(ω) , Lsν (ψ p )
E2
.
ds

i
X h
N ,(ω)
+
E MT
(ψ p )2  .
p≥1
By Lemma 4.5, we have:
D
E
N
N ,(ω)
, Lsν (ψ) ≤ C ηsN ,(ω) ηs
−3,2α
ψ
6,α
.
Then,
Z
T
E
0

XD
p≥1
ηsN ,(ω) ,
Lsν
N
(ψ p )
E2
Z
ds ≤ C 2
T
N ,(ω) 2
E ηs
0
−3,2α
N ,(ω) 2
≤ C T sup E ηs
2
s≤T
−3,2α
ds,
≤ C 2 TAN ,
where AN is defined in Proposition 4.4. The result follows.
Proposition 4.9. 1. For every N , for P-almost every (ω), the trajectories of the fluctuations process
ηN ,(ω) are almost surely continuous in S ′ ,
2. For every N , for P-almost every (ω), the trajectories of M N ,(ω) are almost surely continuous in
S ′.
816
Proof. We only prove for M N ,(ω) , since, using Proposition 4.8, the proof is the same for ηN ,(ω) . Let
−3,2α
(ϕ p ) be a complete orthonormal system in W0
, then for every fixed N and (ω), we know from
the proof of Proposition 4.6, that for all ǫ > 0, there exists some M0 > 0 such that
X
N ,(ω)
sup(M t
p≥M0 t≤T
(ϕ p ))2 <
ǫ
3
, a.s.
Let (t m ) be a sequence in [0, T ] such that t m →m→∞ t.
2
N ,(ω)
N ,(ω) − Mt
Mtm
−3,2α
=
X
N ,(ω)
− Mt
N ,(ω)
− Mt
Mtm
p≥1
≤
M0 X
Mtm
p=1
N ,(ω)
N ,(ω)
2
2
(ϕ p ),
(ϕ p ) +
2ǫ
3
≤ ǫ,
if t m is sufficiently large.
We are now in position to prove the tightness of the fluctuations process. Let us recall some notations: for fixed N and (ω) H N ,(ω) is the law of the process ηN ,(ω) . Hence, H N ,(ω) is an element
of M1 (C ([0, T ], S ′ )), endowed with the topology of weak convergence and with B ∗ , the smallest
σ-algebra such that the evaluations Q 7→ Q , f are measurable, f being measurable and bounded.
We will denote by ΘN the law of the random variable (ω) 7→ H N ,(ω) . The main result of this part is
the following:
Theorem 4.10. 1. for P-almost every sequence (ω), the law of the process M N ,(ω) is tight in
M1 (C ([0, T ], S ′ )),
2. The law of the sequence (ω) 7→ H N ,(ω) is tight on M1 M1 (C ([0, T ], S ′ )) .
Before proving Theorem 4.10, we recall the following result and notations (cf. Mitoma [19], Th 3.1,
p. 993):
Proposition 4.11 (Mitoma’s criterion).
Let (PN ) be a sequence of probability measures on (CS ′ := C ([0, T ], S ′ ), BCS ′ ). For each ϕ ∈ S , we
denote by Πϕ the mapping of CS ′ to C := C ([0, T ], R) defined by
Πϕ : ψ(·) ∈ CS ′ 7→ ψ(·) , ϕ ∈ C .
′
Then, if for all ϕ ∈ S , the sequence (PN Π−1
ϕ ) is tight in C , the sequence (PN ) is tight in CS .
Remark 4.12.
A closer look to the proof of Mitoma shows that it suffices to verify the tightness of (PN Π−1
ϕ ) for ϕ
in a countable dense subset of the nuclear Fréchet space (S , k · k p , p ≥ 1).
Thanks to Mitoma’s result, it suffices to have a tightness criterion in R. We recall here the usual result
(cf. Billingsley [4]): A sequence of (ΩN , F tN )-adapted processes (Y N ) with paths in C ([0, T ], R) is
tight if both of the following conditions hold:
817
• Condition [T]: for all t ≤ T and δ > 0, there exists C > 0 such that
€
Š
sup P |YtN | > C ≤ δ,
N
(Tt,δ,C )
• Condition [A]: for all η1 , η2 > 0, there exists C > 0 and N0 such that for all F N -stopping
times τN ,
‹

N
N
(Aη1 ,η2 ,C )
sup sup P Yτ − Yτ +θ ≥ η2 ≤ η1 .
N
N
N ≥N0 θ ≤C
Proof of Theorem 4.10.
1. Tightness of (M N ,(ω) ): for a fixed realization of the disorder (ω), for
fixed ϕ ∈ S , we have:
• For all t ∈ [0, T ], for all δ > 0, for all C > 0,
h
n
oi
‹ E sup t≤T M tN ,(ω) (ϕ)2

N ,(ω)
(ϕ) > C ≤
P M t
,
C2
2
N ,(ω) 2
E sup t≤T M t
ϕ 3,2α
−3,2α
≤
,
C2
2
N €
C ϕ 3,2α
Š
1X
4α
1
+
|ω
|
, (cf. (31)),
sup
≤
i
a2
N N i=1
≤ δ,
for a suitable C > 0 (depending on (ω)). Condition [T] is proved.
• Let us verify Condition [A]: For every ϕ ∈ S , for every δ, θ , η1 , η2 > 0, θ ≤ δ, for every
stopping time τN ,

‹
2 1
N
N
N
N
uN := P Mτ +θ (ϕ) − Mτ (ϕ) > η2 ≤ 2 E Mτ +θ (ϕ) − Mτ (ϕ) ,
N
N
N
N
η
Z
 2
τN +θ
¬
¶
1
νsN , ϕ ′ ( y, π)2 ds ,
≤ 2E
η2
τN

Z
τN +θ Z
2
2 1
N
H 
≤ ϕ 6,α 2 E 
y,π −6,α dνs ds ,
η2
τN
S1 ×R

Z
τN +θ
N
2 C
1X
≤ ϕ 6,α 2 E 
(1 + |ωi |4α ) ds , (cf. (18) and (21)),
N
η2
τN
i=1
!
N
2 Cδ
1X
4α
(1 + |ωi | ) .
≤ ϕ 6,α 2 sup
N i=1
η2 N
This last term is lower or equal than η1 for δ sufficiently small (depending on (ω)).
2. Tightness of (ΘN ): we need to be more careful here, since the tightness is in law w.r.t. the
disorder. Let (ϕ j ) j≥1 be a countable family in the nuclear Fréchet space S . Without any
818
restriction, we can always suppose that φ j 6,α = 1, for every j ≥ 1. We define the following
decreasing sequences (indexed by J ≥ 1) of subsets of M1 C ([0, T ], S ′ ) :
n
o
K1ǫ (ϕ1 , . . . , ϕJ ) := P ; ∀t, ∀1 ≤ j ≤ J, PΠ−1
satisfies
(T
)
,
t,δ,C
ϕj
1
n
o
K2ǫ (ϕ1 , . . . , ϕJ ) := P ; ∀1 ≤ j ≤ J, ∀η1 , η2 > 0, PΠ−1
satisfies
(A
)
,
η1 ,η2 ,C2
ϕ
j
where C1 = C1 (ǫ, δ), C2 = C2 (ǫ, η1 , η2 ) will be precised later. By construction and by Mitoma’s
theorem (cf. Remark 4.12),
\
K ǫ := (K1ǫ (ϕ1 , . . . , ϕJ ) ∩ K2ǫ (ϕ1 , . . . , ϕJ ))
J
is a relatively compact subset of M1 C ([0, T ], S ′ ) . In order to prove tightness of (ΘN ), it
is sufficient to prove that, for all ǫ > 0,
!
[
ǫ
c
Ki (φ1 , . . . , φJ ) ≤ ǫ.
(33)
∀i = 1, 2, lim sup ΘN
N
J
For ǫ > 0, let A = A(ǫ) such that lim infN →∞ P AN ≤ A ≥ 1 − ǫ, and
!
N
1X
lim inf P
(1 + |ωi |4α ) + AN (ω1 , . . . , ωN ) ≤ A ≥ 1 − ǫ,
N →∞
N i=1
where AN is the random variable defined in Proposition 4.4. We define the corresponding
constants (for a sufficiently large constant C):
r
η1 η22
A(ǫ)
C1 (ǫ, δ) :=
, C2 (ǫ, η1 , η2 ) :=
.
δ
CA(ǫ)
Then,

ΘN (K1ǫ (φ1 , . . . , φJ ))


≥ P (ω), ∀t, ∀1 ≤ j ≤ J, ∀δ,

‚
N ,(ω) 2
≥ P (ω), sup E η t
−6,α
t≤T
D
E 2 N ,(ω)
, φj E ηt
C1 (δ, ǫ)2



≤ δ ,

Œ
≤ A , (by definition of C1 ),
≥ P AN ≤ A(ǫ) , (cf. (18) and (25)).
S
Letting J → ∞ in the latter inequality, we obtain: ΘN ( J K1ǫ (φ1 , . . . , φJ )c ) ≤ P(AN > A).
Taking on both sides lim supN →∞ , we get the result.
819
Furthermore, for η2 > 0, 0 < θ ≤ C2 and τN ≤ T a stopping time, for all 1 ≤ j ≤ J,

Z
2 
!
τN +θ D
Z τN +θ D
E
E
1
N
N


P ηsN ,(ω) , Lsν (ϕ j ) ds ≥ η2 ≤ 2 E 
ηsN ,(ω) , Lsν (ϕ j ) ds  ,
τ
η2
τN
N

Z
τN +θ D
E 2
C2
N
≤ 2E
ηsN ,(ω) , Lsν (ϕ j ) ds ,
η2
τN
Z T D
E2
C2
N
≤ 2
E ηsN ,(ω) , Lsν (ϕ j ) ds,
η2 0
Z T h
i
2
C C2
≤ 2
E ηsN −3,2α ds,
η2 0
C T C2
AN , (cf. (25)).
≤
η22
And,
‹ CC

N
2
N
P Mτ +θ (ϕ j ) − Mτ (ϕ j ) > η2 ≤ 2
N
N
η2
N
1X
N
i=1
!
4α
(1 + |ωi | )
.
So, for all j ≥ 1, by definition of C2 ,

‹
η1
N ,(ω)
P ητ +θ (ϕ j ) − ητN ,(ω) (ϕ j ) ≥ η2 ≤
N
N
A(ǫ)
AN +
N
1X
N
i=1
!
4α
(1 + |ωi | )
.
Consequently,
ΘN (K2ǫ (φ1 , . . . , ϕJ )) ≥ P
Letting J → ∞, we get lim supN ΘN
4.3.3
€S
J
AN +
N
1X
N
i=1
!
(1 + |ωi |4α ) > A(ǫ)
.
Š
K2ǫ (ϕ1 , . . . , ϕJ )c ≤ ǫ. Eq. (33) is proved.
Identification of the limit
The proof of the fluctuations result will be complete when we identify any possible limit.
Proposition 4.13 (Identification
of theinitial value).
N ,(ω)
The random variable (ω) 7→ L η0
converges in law to the random variable ω 7→ L (X (ω)),
where for all ω, X (ω) = C(ω) + Y , with Y a centered Gaussian process with covariance Γ1 . Moreover
ω 7→ C(ω) is a Gaussian process with covariance Γ2 , where Γ1 and Γ2 are defined in (8) and (9).
D
E
N ,(ω)
Proof. For simplicity, we only identify here the law of η0
, ϕ for all ϕ. The same proof works
E
E
D
D
N ,(ω)
N ,(ω)
, ϕ p , p ≥ 1. We write
for the law of finite-dimensional distributions
η0
, ϕ1 , . . . , η0
820
Γi for Γi (ϕ, ϕ), i = 1, 2. One has:
D
N ,(ω)
η0
,
‚
Œ
Z
N
1 X
i
ϕ(ξ , ωi ) −
ϕ(x, ωi )λ( dx)
ϕ =p
N i=1
S1
‚Z
Œ
N
1 X
+p
ϕ(x, ωi )λ( dx) − ν0 , ϕ ,
N i=1
S1
E
=: AN ,(ω) + B N ,(ω) .
It is easy to see that B N ,(ω) converges in law to Z2 ∼ N (0, Γ2 ). Moreover, for P-almost every (ω),
AN ,(ω) converges in law to Z1 ∼ N (0, Γ1 ) (see Billingsley [5], Th. 27.3 p. 362). That means that for
u2
− 2Γ
iuAN ,(ω)
all u ∈ R, ψAN (u) := Eλ e
converges to ψY (u) := e 1 . But, then, for all F ∈ C b (R),
D N ,(ω) E h h
ii
h i
N ,(ω)
N ,(ω)
,ϕ
iu η0
+B N ,(ω) )
= E F Eλ e iu(A
= E F e iuB
ψN (u) .
E F Eλ e
Since ψN (u) converges almost
to a constant, the limit of the expression above exists (Slutsky’s
surely
2
theorem) and is equal to E F
e
u
iuZ2 − 2Γ
1
.
Proposition 4.14 (Identification of the martingale part).
For P-almost every (ω), the sequence (M N ,(ω) ) converges in law in C ([0, T ], S ′ ) to a Gaussian process
W with covariance defined in (7).
Proof. For fixed (ω), (M N ,(ω) ) is a sequence of uniformly square-integrable continuous martingales (cf. Remark 4.7), which is tight in C ([0, T ], S ′ ). Let W1 and W2 be two accumulation
points (continuous square-integrable martingales which a priori depend on (ω)) and (M φ(N ),(ω) )
and (M ψ(N ),(ω) ) be two subsequences converging to W1 and W¬
2 , respectively. Note that we¶ can
suppose that φ(N ) ≤ ψ(N ) for all N . For all ϕ ∈ S , limN →∞ M φ(N ),(ω) (ϕ) , M ψ(N ),(ω) (ϕ) t =
W1 (ϕ) , W2 (ϕ) t , for all t, and
¬
M
φ(N ),(ω)
(ϕ) , M
ψ(N ),(ω)
(ϕ)
Z
¶
t
t
=
¬
¶
νsφ(N ) , (ϕ ′ )2 ds.
0
We now have to identify the limit: we already know that for P-almost every realization of the
disorder (ω), (ν N ,(ω) ) converges in law
R t ¬to P. But,¶the latter expression, seen as a function of ν, is
continuous. So W1 (ϕ) , W2 (ϕ) t = 0 Ps , (ϕ ′ )2 . So W1 − W2 is a continuous square integrable
martingale whose Doob-Meyer process is 0. So W1 = W2 and is characterized as the Gaussian process
with covariance given in (7). The convergence follows.
Proof of the independence of W and X . We prove more : the triple (Y, C, W ) is independent. For sake
of simplicity, we only consider the case of (Y (ϕ), C(ϕ), Wt (ϕ)) for fixed t and ϕ.
N ,(ω)
Let us first recall some notations: let AN ,(ω) , B N ,(ω) and M t
(ϕ)
therandom variables
beN ,(ω)
defined
N ,(ω)
iuA
in the proof of Proposition 4.13 and 4.14 and let ψAN (u) := E e
, ψB N (v) := E e i vB
,
N ,(ω)
ψ M N (w) := E e iwM t (ϕ) be their characteristic functions (u, v, w ∈ R). We know that, for almost
821
−
u2
toŠ ψY (u) = e 2Γ1 and that
every (ω), ψAN (u) converges
€
€ ψŠM N (w) converges to the deterministic
iwWt (ϕ)
function ψW (w) := E e
. But, if ψC (v) = E e iwC , then, for all u, v, w ∈ R, using the
independence of the Brownian with the initial conditions,
N ,(ω)
N ,(ω)
N ,(ω)
N ,(ω)
(ϕ)
+iwM t
E E e iuA +i vB
− e i vB
ψAN (u)ψ M N (w) = 0.
Using Slutsky’s theorem, we see that any limit couple (Y, C, W ) satisfies
€ €
ŠŠ
E E e iuY +i vC+iwWt (ϕ) = ψY (u)ψC (v)ψW (w).
which is the desired result.
We recall that the limit second order differential operator Ls is defined by
Ls (ϕ)( y, π) :=
1
2
ϕ ′′ ( y, π) + ϕ ′ ( y, π)(b[ y, Ps ] + c( y, π)) + Ps , ϕ ′ (·, ·)b(·, y, π) .
As in Lemma 4.5, we can prove the following:
Lemma 4.15.
N
Assume (H b,c ). Then for every N , s ≤ T , (ω), the operator Ls and Lsν are linear continuous from S
to S and
L (ϕ) ≤ C ϕ ,
s
6,α
8,α
νN
Ls (ϕ) ≤ C ϕ 8,α .
6,α
We are now in position to prove Theorem 2.10:
Proof of Theorem 2.10. Let Θ be an accumulation point of ΘN . Thus, for a certain subsequence
(which will be also denoted as N for notations purpose), the random variable (ω) 7→ H N ,(ω) converges in law to a random variable H with values in M1 (C ([0, T ], S ′ )) with law Θ. Applying
Skorohod’s representation theorem, there exists some probability space (Ω(1) , P(1) , F (1) ) and random variables defined on Ω(1) , ω1 7→ H N (ω1 ) and ω1 7→ H(ω1 ) such that H N has the same law as
(ω) 7→ H N ,(ω) , H has the same law as H , and for P(1) -almost every ω1 ∈ Ω(1) , H N (ω1 ) converges
to H(ω1 ) in M1 (C ([0, T ], S ′ )).
(1)
An easy application of
surely,
Proposition 4.8 and Borel-Cantelli’s Lemma shows that P -almost
Rt
ω ∗ ω1
1 E sup t≤T η t −6,α < ∞. Then we know from Lemma 4.15 that the integral term 0 Ls ηs ds
−8,α
makes sense as a Bochner’s integral in W0
⊆ S ′.
Let ηN ,ω1 with law H N (ω1 ); ηN ,ω1 converges in law to some ηω1 with law H(ω1 ). By uniqueness
ω
in law convergence, using Propositions 4.13 and 4.14, we see that (η0 1 , W ) as the same law as
′
(X (ω1 ), W ). For fixed ϕ ∈ S , we define Fϕ from C ([0, T ], S ) into R by Fϕ (γ) := γ t , ϕ −
Rt
γ0 , ϕ − 0 γs , Ls ϕ ds. The function Fϕ is continuous and since ηN ,ω1 converges in law to ηω1 ,
the sequence (Fϕ (ηN ,ω1 )) converges in law to Fϕ (ηω1 ). To prove the theorem, it remains to show
E
R t D N ,ω
N
that the law of the term 0 ηs 1 , Lsν ϕ − Ls ϕ ds converges in law to 0. We show that there
822
is convergence in probability: For all ǫ > 0, for all A > 0, using Proposition 4.8, Lemma 4.15, and
Cauchy-Schwarz’s inequality,
‚ –Z t ™
Œ
E
D
N ,ω1
(1)
νN
UN ,ǫ := P
E
, (Ls − Ls )(ϕ) ds > ǫ ,
ηs
0
‚ –Z t ™
Œ
E
D
N ,(ω)
νN
=P E
, (Ls − Ls )(ϕ) ds > ǫ ,
ηs
0
‚Z
≤P
t
N ,(ω) 2
E ηs
−6,α
0
‚
≤P
6,α
Z
≤ P CN (ω1 , . . . , ωN )1/2
‚Z
Œ
1/2 2 1/2
N
ds > ǫ ,
E (Lsν − Ls )(ϕ) t
0
Œ
2 1/2
νN
ds > ǫ (cf. Prop 4.8),
E (Ls − Ls )(ϕ) 6,α
Œ
t 2 1/2
ǫ
νN
+ P CN > A .
ds > p
E (Ls − Ls )(ϕ) 6,α
A
0
Using (4.8), it suffices to prove that, for all ǫ > 0,
Œ
‚Z t 2 1/2
νN
ds > ǫ = 0.
lim sup P
E (Ls − Ls )(ϕ) N →∞
6,α
0
(34)
Indeed, for every ϕ ∈ S ,
N
UsN (ϕ)( y, π) := (Lsν − Ls )(ϕ)( y, π) = ϕ ′ ( y, π)(b[ y, νsN ] − b[ y, Ps ]).
An analogous calculation as in Lemma 4.5 shows that, using Lipschitz assumptions on b, and Proposition 4.3:

2 
N
 ≤ ϕ 2 (C/N + D (ω , . . . , ω )),
E 
N
1
N
sup Us (ϕ) 8,α
s≤t
6,α
with the property that limA→∞ lim supN P(N DN > A) = 0. Equation (34) is a direct consequence.
Since there is uniqueness in law in (10), Θ is perfectly defined, and thus, unique. The convergence
follows.
5 Proofs for the fluctuations of the order parameters
We end by the proofs of paragraph 2.3.3.
5.1 Proof of Proposition 2.13
¬
¶
1. This is straightforward since r N ,(ω) = ν N ,(ω) , e i x and since for P-almost every disorder
(ω), ν N ,(ω) converges weakly to P.
823
2. The following sequences are well defined: ∀k ≥ 0,
Z
e−|ω| ωk cos(θ )Pt ( dθ , dω),
uk (t) :=
ZS
1 ×R
e−|ω| ωk sin(θ )Pt ( dθ , dω).
vk (t) :=
S1 ×R
Let E = (ℓ∞ (N), k · k∞ ) be the Banach space of real bounded sequences endowed with its usual
k · k∞ norm, (k u k∞ = supk≥0 |uk |). For all t > 0, let A t : E × E → E × E, be the following
linear operator (where (u, v) is a typical element of E × E): For all k ≥ 0
¨
A t (u, 0)k = − 21 uk − αk (t)v0 + βk (t)u0 − K vk+1 ,
A t (0, v)k = − 21 vk + γk (t)v0 − αk (t)u0 + Kuk+1 ,
where,
¬
¶
αk (t) = Pt , e−|ω| ωk cos(·) sin(·) ,
¬
¶
βk (t) = Pt , e−|ω| ωk sin2 (·) ,
¬
¶
γk (t) = Pt , e−|ω| ωk cos2 (·) .
(t, u, v) 7→ A t · (u, v) is globally Lipschitz-continuous map from [0, T ] × E × E into E × E
and one easily verifies considering (4) (in the case of the sine-model) and developing the
sine interaction that t 7→ (u(t), v(t)) satisfies in E × E the following linear non-homogeneous
Cauchy Problem:

d
· (u(t), v(t)),
 dt (u(t), v(t)) = A
¬ t
¶
uk (0)
= ¬ P0 , e−|ω| ωk cos(·)¶ , ∀k ≥ 0

vk (0)
=
P0 , e−|ω| ωk sin(·) , ∀k ≥ 0.
Let us suppose that there exists some t 0 ∈ [0, T ] such that r t 0 = 0, namely u0 (t 0 ) = v0 (t 0 ) = 0.
Then, if (ũ, ṽ) is the constant function on [0, T ] such that for all k ≥ 0, ũk ≡ uk (t 0 ), ṽk ≡ vk (t 0 ),
then (ũ, ṽ) satisfy the same Cauchy Problem as (u, v) with initial condition at time t 0 . By
Cauchy-Lipschitz theorem, both functions coincide on [0, T ]. In particular, u0 and v0 are
always zero and thus r ≡ 0.
3. We suppose (H r ). A simple calculation shows that the fluctuations process R N verifies for all
t ∈ [0, T ],
D
ED
E D
ED
E
N ,(ω)
N ,(ω)
N ,(ω)
N ,(ω)
ηt
, cos(·) ν t
+ Pt , cos(·) + η t
, sin(·) ν t
+ Pt , sin(·)
N ,(ω)
Rt
=
,
N ,(ω)
rt
+ rt
D
E
ED
N ,(ω)
N ,(ω)
ix
i
x
νt
+ Pt , e
ℜ
ηt
,e
.
=
D
E
N ,(ω) i x , e + rt
νt
¬
¶
¬
¶
¬
¶
¬
¶
Let uN ,(ω) := ν N ,(ω) , e i x , v N ,(ω) := ηN ,(ω) , e i x and u := P , e i x , v ω := ηω , e i x
be their corresponding limits. The result follows if we prove the following property:
824
€
Š
the random variables (ω) 7→ L uN ,(ω) , v N ,(ω) converges in law to the random variable
ω 7→ L (u, v ω ). The tightness of this random variable follows from the convergence
of both empirical measure and fluctuations process. As already said in Remark 2.14,
N ,(ω) N ,(ω)
, vt
) =
it suffices to prove the convergence of the finite-dimensional marginals (u t
N ,(ω)
N ,(ω)
N ,(ω)
N ,(ω)
(u t 1 , . . . , u t p ), (vt 1 , . . . , vt p ) , for all element of [0, T ], t 1 , . . . , t p , p ≥ 1.
Since the limit of (uN ,(ω) ) is a constant, this is mainly a consequence of Slutsky’s theorem. But
since this is a convergence in law with respect
to the
the proof. We
€
€ has to adapt
Š disorder, one
Š
prove the following: ∀G ∈ C b1 (R), ∀r = r1 , . . . , r p ∈ R p , ∀s = s1 , . . . , s p ∈ R p ,
‹˜
• 
h i
→ E G ϕ(u t ,v ωt ) (r, s) ,
E G ϕ(uN ,(ω) ,v N ,(ω) ) (r, s)
t
N →∞
t
”
—
where ϕ(X ,Y ) (r, s) = E e i r·X +is·Y is the characteristic function of the couple (X , Y ). Indeed,
we have successively:
• 
‹˜
h i
− E G ϕ(u t ,v ωt ) (r, s) ,
aN := E G ϕ(uN ,(ω) ,v N ,(ω) ) (r, s)
t
t
• 
‹˜
≤ E G ϕ(uN ,(ω) ,v N ,(ω) ) (r, s)
− E G ϕ(u ,v N ,(ω) ) (r, s) t
t
t t
h i
+ E G ϕ(u ,v N ,(ω) ) (r, s)
− E G ϕ(u t ,v ωt ) (r, s) ,
t t
≤ C E ϕ(uN ,(ω) ,v N ,(ω) ) (r, s) − ϕ(u ,v N ,(ω) ) (r, s)
t
t
t t
h i
− E G ϕ(u t ,v ωt ) (r, s) ,
+ E G ϕ(u ,v N ,(ω) ) (r, s)
t t
N ,(ω)
i r·u ≤ C EE e i r·u t
− e t
h i
− E G ϕ(u t ,v ωt ) (r, s) .
+ E G ϕ(u ,v N ,(ω) ) (r, s)
t
t
i r ·uN ,(ω)
i
r·u
N ,(ω)
− e t ≤ min 2, |r||u t
− u t | . So, for all ǫ > 0,
But, we have E e t t

‹
N ,(ω)
N ,(ω)
i r·u E e i r·u t
− e t ≤ ǫ|r| + 2P u t
− ut > ǫ .
Taking lim supN →∞ , and letting ǫ → 0, we get lim aN = 0. The result follows.
5.2 Proof of Proposition 2.15
The proof is similar to the previous one and relies on the two following equalities :
¬
¶
ix
P
,
e
,
ζN ,(ω) =
r N ,(ω)
€ ¬
¶ ¬
¶
Š
Š
p € N ,(ω)
1
N ,(ω)
ix
ix
N ,(ω)
r
η
,
e
+
P
,
e
R
.
N ζ
−ζ =
r · r N ,(ω)
825
A Proof of Proposition 4.3
Thanks to the Lipschitz continuity of b and c, introducing ν as the empirical measure corresponding
ω
ω
to (x ωi , ωi ), we have, (inserting b[x s i , νsN ] − b[x s i , νs ] in the b term),
–
™
2
E sup x si,N − x sωi ‚Z
≤C
s≤t
t
E
0
Z
t
+
E
h€
0
‚ Z
≤C
Z
0
t
+
E
0
h€
b[x si,N , νsN ] − b[x sωi , Ps ]
c(x si,N , ωi ) − c(x sωi , ωi )
t
2
h€
–
Š2 i
Š2 i
Œ
ds ,
™
2
E sup x ui,N − x uωi Z
t
ds +
u≤s
ds
2 ™
ωj
j,N sup E sup x u − x u ds
–
u≤s
0 1≤ j≤N
b[x sωi , νs ] − b[x sωi , Ps ]
Š2 i
Œ
ds .
2 ωj
j,N Applying Gronwall’s Lemma to sup1≤ j≤N E supu≤t x u − x u , it suffices to prove that for some
ZN :
Z
t
E
h€
0
b[x sωi , νs ] − b[x sωi , Ps ]
Š2 i
ds ≤ C/N + ZN (ω1 , . . . , ωN ).
Indeed, for all 1 ≤ i ≤ N , (we write x i instead of x ωi to simplify notations):


N
X
X
€
Š2
1 

ui,N := b[x si , νs ] − b[x si , Ps ] = 2 
T (x i , x j )2 +
T (x i , x k )T (x i , x l ) ,
N
j=1
k6=l
R
j
where T (x i , x j ) := b(x si , x s , ω j ) − b(x si , y, π)Ps ( d y, dπ). Since b is bounded, we see that the first
term is of order (1/N ). We only have to study the remaining term:




X

 X
i
k
i
l 
T
(x
,
x
)T
(x
,
x
)
E  T (x i , x k )T (x i , x l ) ≤ C N + E 
.

k6=i,l6=i
k6=l
k6=l
Since the (x i ) are independent, if we take conditional expectation w.r.t. (x r , r 6= l) in the last term,
we get:


˜˜
• •X
X


i
k
i
l r
i
k
i
l 
x
,
r
=
6
l
,
T
(x
,
x
)T
(x
,
x
)
=
E
E
T
(x
,
x
)T
(x
,
x
)
E


k6=i,l6=i
k6=l





 X

 X
i
i 
i
k
i 

G
(x
)G
(x
)
=
E
T
(x
,
x
)G
(x
)
= E
k
l
l
,



k6=i,l6=i
k6=i,l6=i
k6=l
k6=l
826
where Gl (x) = G(x, ωl ) =
R
ω
b(x, y, ωl )Ps l ( d y) −
Z
C
ZN (ω1 , . . . , ωN ) :=
N
T
0
R
b(x, y, π)Ps ( d y, dπ). Defining

N
1 X

E p
G(x si , ωl )
N l=1
!2 

 ds,
in order to prove (24) it suffices to show that for some constant C,


!2  
Z T
N
X
1
 


G(x si , ωl )  ds ≤ C.
E p
E
N l=1
0
The rest of the proof
to prove this last assertion: we have successively (setting
PN is devoted
ωi
ωi
1
p
UN (x s , ω) := N l=1 G(x s , ωl ))
Z

T
E
E
”
UN (x sωi , ω)2
0
—
Z
ds ≤
≤
≤
+
T
” ”
——
E E UN (x sωi , ω)2
ds,
0
1
Z
E E 
N
1
0
Z
N
Z
1
N
 
T
N
N X
X

G(x sωi , ωk )G(x sωi , ωl ) ds,
k=1 l=1
T
 
E E 
0
T
N
X

G(x sωi , ωl )2 
ds + C
l=1

 
E E 
0
X
G(x sωi , ωk )G(x sωi , ωl ) ds.
l6=k, l6=i, k6=i
The first term of the RHS of the last inequality is bounded, since b is bounded. But, if we condition
w.r.t. ω r for r 6= i, r 6= k, we see that the second term is zero. The result follows.
Acknowledgements
This is a part of my PhD thesis. I would like to thank my PhD supervisors Giambattista Giacomin and
Lorenzo Zambotti for introducing this subject, for their useful advice, and for their encouragement.
I would also like to thank the referee for useful suggestions and comments.
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